ARS needs to include both the design and random variables, the number of

variables is relatively large, often exceeding 15 variables. This excludes the use of many

DOE, such as Central Composite Design (CCD). The CCD has 2n vertices, 2n axial

points, and one center point, so the required number of design points is 2n+2n+1,where n

is the number of variables involved. A polynomial of nmth order in terms of n variables has

L coefficients, where

 (n + 1)(n + 2)...(n + m)
 L = (3-7)


For n = 15, CCD requires 32799 analyses. On the other hand, a quadratic polynomial in

15-variable has 136 coefficients. From our experience, in order to estimate these

coefficients, the number of analyses only needs to be about twice as large as the number

of coefficients, which is less than one percent of the number of vertices for 15-variable

space. Therefore, other DOEs such as CCD using fractional factorial design (Myers and

Montgomery 1995) need to be used. The fractional factorial CCD is intended for the

construction of quadratic RSA. Orthogonal arrays (Myers and Montgomery 1995) are

used for the construction of higher order RS (Balabanov 1997 and Padmanabhan et al.

2000). Isukapalli (1999) employed orthogonal arrays to construct SRS. For problems

where only very limited number of analyses is computationally affordable, Box-Behnken

designs or saturated designs can be used (Khuri and Cornell 1996).

 In the paper of Qu et al. (2000), it is shown that Latin Hypercube sampling is more

efficient and flexible than orthogonal arrays. The idea of Latin Hypercube sampling can

be explained as follows: assume that we want n samples out of k random variables. First,

the range of each random variable is divided into n nonoverlapping intervals on the basis

of equal probability. Then one value is selected randomly from each interval. Finally, by